Adaptive control system having direct output feedback and related apparatuses and methods

ABSTRACT

An adaptive control system (ACS) uses direct output feedback to control a plant. The ACS uses direct adaptive output feedback control developed for highly uncertain nonlinear systems, that does not rely on state estimation. The approach is also applicable to systems of unknown, but bounded dimension, whose output has known, but otherwise arbitrary relative degree. This includes systems with both parameter uncertainty and unmodeled dynamics. The result is achieved by extending the universal function approximation property of linearly parameterized neural networks to model unknown system dynamics from input/output data. The network weight adaptation rule is derived from Lyapunov stability analysis, and guarantees that the adapted weight errors and the tracking error are bounded.

CROSS-REFERENCE TO RELATED APPLICATIONS

This continuation application claims priority benefits of U.S.provisional application No. 60/208,101 filed May 27, 2000 andnonprovisional application Ser. No. 09/865,659 filed May 25, 2001 namingAnthony J. Calise, Naira Hovakimyan, and Hungu Lee as inventors

FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was funded in part by the Air Force Office of ScientificResearch (AFOSR) under Grant No. F4960-01-1-0024. The United StatesGovernment therefore has certain rights in the invention.

FIELD OF INVENTION

The invention is directed to a system, apparatuses and methods foradaptively controlling a plant such an aircraft, automobile, robot, orother controlled system.

BACKGROUND OF THE INVENTION

Research in adaptive output feedback control of uncertain nonlineardynamic systems is motivated by the many emerging applications thatemploy novel actuation devices for active control of flexiblestructures, fluid flows and combustion processes. These include suchdevices as piezoelectric films, and synthetic jets, which are typicallynonlinearly coupled to the dynamics of the processes they are intendedto control. Modeling for these applications vary from having accuratelow frequency models in the case of structural control problems, tohaving no reasonable set of model equations in the case of activecontrol of flows and combustion processes. Regardless of the extent ofthe model accuracy that may be present, an important aspect in anycontrol design is the effect of parametric uncertainty and unmodeleddynamics. While it can be said the issue of parametric uncertainty isaddressed within the context of adaptive control, very little can besaid regarding robustness of the adaptive process to unmodeled internalprocess dynamics.

Synthesis approaches to adaptive output feedback control typically makeuse of state estimation, and therefore require that the dimension of theplant is known. Some approaches further restrict the output to have fullrelative degree, or restrict the uncertainties in the plant to be anunknown function of the output variables. It would be desirable toremove all these restrictions by adopting a direct output feedbackapproach that does not rely on state estimation. One of the immediateconsequences of such an approach would be that the dimension of thecontrolled plant need not be known. Consequently, the resulting systemwould be applicable to plants having both parametric uncertainty andunmodeled dynamics. Furthermore, it would be desirable to produce acontrol system that is not only robust to unmodeled dynamics, but alsolearns to interact with and control these dynamics.

Output feedback control of full relative degree systems was introducedby Esfandiari and Khalil, 1992, “Output feedback stabilization of fullylinearizable systems,” International Journal of Control,56(5):1007-1037. In their publication the authors formulated a controlmethodology that involves a high gain observer for the reconstruction ofthe unavailable states. A solution to the output feedback stabilizationproblem for systems in which nonlinearities depend only upon theavailable measurement, was given by Praly, L. and Jiang, Z. (1993),“Stabilization by output feedback for systems with iss inversedynamics,” System & Control Letters, 21:19-33. Krstic, M.,Kanellakopoulos, I., and Kokotovic, P. (1995), Nonlinear and AdaptiveControl Design, John Wiley & Sons, Inc. New York and Marino, R. andTomei, P. (1995). Nonlinear Control Design: Geometric, Adaptive, &Robust. Prentice Hall, Inc., presented backstepping-based approaches toadaptive output feedback control of uncertain systems, linear withrespect to unknown parameters. An extension of these methods due toJiang can be found in Jiang, Z. (1999), A combined backstepping andsmall-gain approach to adaptive output feedback control. Automatica,35:1131-1139.

For adaptive observer design, the condition of linear dependence uponunknown parameters has been relaxed by introducing a neural network (NN)in the observer structure of Kim, Y. and Lewis, F. (1998), High LevelFeedback Control with Neural Networks, World Scientific, N.J. Adaptiveoutput feedback control using a high gain observer and radial basisfunction neural networks (NNs) has also been proposed by Seshagiri, S.and Khalil, H. (2000), “Output feedback control of nonlinear systemsusing {RBF} neural networks,” IEEE Transactions on Neural Networks,11(1):69-79 for nonlinear systems, represented by input-output models.Another method that involves design of an adaptive observer usingfunction approximators and backstepping control can be found in Choi, J.and Farrell, J. (2000), “Observer-based backstepping control usingon-line approximation,” Proceedings of the American Control Conference,pages 3646-3650. However, this result is limited to systems that can betransformed to output feedback form, i.e., in which nonlinearitiesdepend upon measurement only.

The state estimation based adaptive output feedback control designprocedure in the Kim and Lewis 1998 publication is developed for systemsof the form:{umlaut over (x)}=f(x)+g(x)δ_(c)  (1)y=x dim x=dim y=dim u,  (2)which implies that the relative degree of y is 2. In Hovakimyan, N.,Nardi, F., Calise, A., and Lee, H. (1999), “Adaptive output feedbackcontrol of a class of nonlinear systems using neural networks,”International Journal of Control that methodology is extended to fullvector relative degree MIMO systems, non-affine in control, assumingeach of the outputs has relative degree less or equal to 2:{umlaut over (x)}=f(x, δ _(c))  (3)y=h(x) dim y=dim u≦dim x.  (4)

These restrictions are related to the form of the observer used in thedesign procedure. Constructing a suitable observer for a highlynonlinear and uncertain plant is not an obvious task in general.Therefore, a solution to adaptive output feedback control problem thatavoids state estimation is highly desirable.

BRIEF SUMMARY OF THE INVENTION

The adaptive control system (ACS) and method of this invention usesdirect adaptive output feedback to control a plant. The system cancomprise a linear controller (LC) and an adaptive element (AE). Thelinear controller can be used as a dynamic compensator to stabilize amodel of the plant, and provide output regulation. The adaptive elementcan compensate for disturbances, and modeling error resulting fromapproximation in modeling of the plant. The adaptive element cancomprise a neural network (NN). The adaptive element can receive asignal from the linear controller used to adapt its NN's weights. Theinput vector to the NN can comprise current and/or past plant outputsignals together with other available signals. The past plant outputsignal(s) can be used as inputs to the NN to ensure boundedness of theadaptive element in controlling the plant. The adaptive control systemcan comprise an error conditioning element having a low-pass filterdesigned to satisfy a strictly positive real (SPR) condition of atransfer function associated with Lyapunov stability analysis of thecontrol system. The stability analysis can be used to construct the NNadaptation law using only the plant output signal(s) and other availablesignals as inputs to the NN, and to ensure boundedness of errorsignal(s) of the closed-loop adaptive control system. Apparatusesforming components of the ACS are also disclosed.

A method of the invention comprises generating at least one controlsignal δ_(c) to regulate a plant output signal y by feedback of theplant output signal y, and optionally other sensed variables related tothe state of the plant, in which y is a function of the plant statehaving known but unrestricted relative degree r. The control signalδ_(c) can be generated so as to control the plant based on anapproximate dynamic model, and so as to control the plant in thepresence of unmodeled dynamics in the plant based on an adaptive controltechnique. The adaptive control technique can be implemented with aneural network. Related methods are also disclosed.

These together with other objects and advantages, which will becomesubsequently apparent, reside in the details of construction andoperation of the invented system, methods, and apparatuses as more fullyhereinafter described and claimed, reference being made to theaccompanying drawings, forming a part hereof, wherein like numeralsrefer to like parts throughout the several views.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general block diagram of an adaptive control system forcontrolling a plant based on a plant output signal y in accordance withthe invention;

FIG. 2 is a schematic diagram of the adaptive control system reduced toelements relevant to tracking error signal analysis;

FIG. 3 is a relatively detailed view of a linear dynamic compensator ofthe adaptive control system;

FIG. 4 is a relatively detailed view of an adaptive element;

FIG. 5 is a block diagram of a processor-based adaptive control systemusing direct output feedback;

FIG. 6 is a flowchart of a general method of the invention;

FIG. 7 is a graph of commanded output signal y_(c) (broken line) andplant output signal y (solid line) versus time for a control systemwithout unmodelled dynamics using only a linear compensator to control aplant;

FIG. 8A is a graph of commanded output signal y_(c) (broken line) andplant output signal y (solid line) versus time for a control systemwithout unmodelled dynamics and with the adaptive element;

FIG. 8B is a graph of adaptive control signal v_(ad) and inversion errorsignal A for a control system without unmodelled dynamics and with theadaptive element;

FIG. 9 is a graph of commanded output signal y_(c) (broken line) andplant output signal y (solid line) versus time for a control system withunmodelled dynamics and with the adaptive element;

FIG. 10A is a graph of commanded output signal y_(c) (broken line) andplant output signal y (solid line) versus time for a control system withunmodelled dynamics, and with the adaptive element; and with the linearcontroller.

FIG. 10B is a graph of commanded output signal y_(c) (heavy line), plantoutput signal y (line with relatively moderate oscillations), andconnection weights W (line with relatively heavy oscillations) versustime for a control system with unmodelled dynamics and with an adaptiveneural network element and linear controller.

DETAILED DESCRIPTION OF THE INVENTION

As used herein, the following terms have the following definitions:

“Actuator” can be virtually any device capable of affecting the state ofa plant to control one or more degrees of freedom thereof. Such actuatorcan be a motor, motor-driven screw, a hydraulic cylinder, a pump orvalve controlling a stream of air, a thermal heater, a compressor orsuction generator, or other device.

“Adaptive control system” means a control system having the capabilityto adapt to changes in a controlled plant or its environment over time.

“And/or” means either one or both of the elements before and after thisterm. Hence, “A and/or B” means “A” or “B” or “A and B”.

“Direct output feedback” refers to a control system, apparatus or methodthat employs feedback of an “output” that is a function of the fullstate(s) existing in a plant. “Direct” refers to the fact that no stateestimation is used to estimate plant states not present in or notderivable from the “output”.

“Operator” can be a human or computer, that receives and input andgenerates and output based on the current and past history of the input,for example, senses a plant output using a plant output signal, andgenerates a commanded state signal to control the plant.

“Memory” can be a random-access memory (RAM), read-only memory (ROM),erasable read-only programmable memory (EPROM), or other memory devicecapable of storing a control program and data executable by a processor.

“Plant” refers to a system controlled by a control system. For example,the plant can be an aircraft, spacecraft, space-launch vehicle,satellite, missile, guided munition, automobile, or other vehicle. Theplant can also be a robot, or a pointing or orientation system such as asatellite orientation system to orient power-generation panels, atransceiver, or a docking mechanism. Such plant can also be a brakingsystem, an engine, a transmission, or an active suspension, or othervehicle subsystem. The plant can be a manufacturing facility or a powergeneration facility. In general, the plant could be virtually anycontrollable system.

“Processor” can be a microprocessor such as a Xeon® or Pentium® brandmicroprocessor produced by Intel® Corporation, an Athlon® brandmicroprocessor commercially available from AMD® Corporation, Sunnyvale,Calif., which can operate at one (1) megahertz or more, amicrocontroller, a field programmable gate array (“FPGA”), aprogrammable logic array (“PLA”), a programmed array logic (“PAL”), orother type of data processing or computing device.

“Relative degree” applies to a regulated variable (such as plant outputsignal y) and corresponds to the number of times the variable must bedifferentiated with respect to time before an explicit dependence on thecontrol variable (such as the command control signal δ_(c)) is revealed.

“Sensor” can be virtually any device(s) for sensing a degree of freedomof a plant's state, whether alone or in combination with one or moreother sensors. The sensor can be virtually any device suitable forsensing information regarding a plant's state. For example, the sensorcould be a gyroscope for detecting orientation of a vehicle such as anaircraft, i.e., pitch or roll attitudes or side slip. The sensor canalso be a temperature or pressure sensor, a position, velocity, orinertial sensor.

“(s)” means one or more of the thing meant by the word preceding “(s)”.Thus, basis function(s) means one or more basis functions.

“State” refers to a property of a plant to be controlled which issufficient to completely define the condition of the plant at any timeinstant. For example, elements of the state can be a position, velocity,acceleration, mass, energy, temperature, pressure, volume, etc. of anobject associated with a plant that is to be controlled.

“State feedback” pertains to a situation in which the entire state ofthe plant can be sensed and used to control the plant through feedback.

“Strictly positive real” is a property that pertains to the transferfunction of a linear time-invariant system. The transfer function, G(s),is a ratio of polynomials in the variable, ‘s’, which is a complexvariable having a real and imaginary part. Let s=a+jb, were a is thereal part and b is the imaginary part. Then the transfer function iscalled ‘Strictly Positive Real’ if the following two conditions aresatisfied:

-   -   1) G(s) is asymptotically stable (all the poles of G(s) have        real parts <0); and    -   2) G(jb)>0 for all values of the real variable ‘b’. This        definition can be found in Khalil, H. K., “Nonlinear Systems,        Second Edition, Prentice-Hall, 1996, p. 404.

“Variable” refers to any signal that can be changed independently of theplant states, such as the control variable, or that are dependent upontime either directly, or indirectly because it depends upon plant statesthat are time varying, such as the output variable.

The present inventions now will be described more fully hereinafter withreference to the accompanying drawings, in which some, but not allembodiments of the inventions are shown. Indeed, these inventions may beembodied in many different forms and should not be construed as limitedto the embodiments set forth herein; rather, these embodiments areprovided so that this disclosure will satisfy applicable legalrequirements. Like numbers refer to like elements throughout.

Many modifications and other embodiments of the inventions set forthherein will come to mind to one skilled in the art to which theseinventions pertain having the benefit of the teachings presented in theforegoing descriptions and the associated drawings. Therefore, it is tobe understood that the inventions are not to be limited to the specificembodiments disclosed and that modifications and other embodiments areintended to be included within the scope of the appended claims.Although specific terms are employed herein, they are used in a genericand descriptive sense only and not for purposes of limitation.

1. General Description of Adaptive Control System and Method

As shown in FIG. 1, an adaptive control system (ACS) 10 can be used tocontrol a plant 12 using ‘direct output feedback’ as opposed to ‘statefeedback.’ In contrast, the disclosed adaptive control system 10 uses‘direct output feedback’ in which a function of the full plant state, asopposed to all plant states or estimates thereof, is used for feedbackcontrol of the plant 12. The adaptive control system 10 of thisinvention is therefore highly useful in the control of plants,especially non-linear plants in which the full state of the plant cannotbe sensed through practical or economically-feasible devices ortechniques, particularly if the plant contains unmodeled or unknowndynamics.

In FIG. 1, the ACS 10 comprises a linear controller (LC) 14, stableadaptive element (AE) 16, and model inversion unit (MIU) 18. The ACS 10can further comprise error signal generator (ESG) 20 and summing unit22. The ACS 10 can further comprise an operator 20, operator interfaceunit 22, and command filter unit 24. These elements permit the operator22, which can be human or machine, to interact with ACS 10 to controlthe plant 18. The plant 12 comprises a system 30 to be controlled. Ingeneral, the system 30 is a non-linear system, although it can belinear. The linear controller 14 is implemented to affect approximatecontrol of the plant, optionally using linear control. The AE 16 servesto implement adaptive control of nonlinearities of the plant 12 that arenot compensated by the linear controller 14. The combined effect of theLC 14 and AE 16 is used to control the plant 12.

The ACS 10 is now described in further detail. The operator interfaceunit 26 is coupled to receive the plant output signal y which, aspreviously described, is a function of the full state of plant 12 havingknown but unrestricted relative degree r in which r is the number oftimes the plant output signal y must be differentiated in order for thecorresponding control variable, δ_(c), to become explicit. The operatorinterface unit 26 can be an electronic interface between an ACS bus andthe operator 24 if implemented as a processor, for example, or can be adisplay, gauge, meter, light, or other indicator if the operator 24 ishuman. The operator 24 generates command signals based on the plantoutput signal y from the operator interface unit 26. The operator 24generates a command action or command signal based on the plant outputsignal y. The operator 24 supplies the command action or command signalto the command filter unit 28 that generates filtered commanded signalsy_(c) and y_(c) ^((r)) based thereon. The integer r denotes the relativedegree of the regulated variable, and signal y_(c) ^((r)) denotes ther^(th) derivative with respect to time of the filtered commanded signaly_(c). The command filter unit 28 is coupled to supply the filteredcommanded signal y_(c) to the error signal generator 20. The errorsignal generator 20 is also coupled to receive the plant output signaly. Based on the filtered commanded signal y_(c) and the plant outputsignal y, the error signal generator 20 generates a tracking errorsignal {tilde over (y)}. The error signal generator 20 is coupled tosupply the tracking error signal {tilde over (y)} to the linearcontroller 14.

The linear controller 14 generates a pseudo-control component signalv_(dc) based on the tracking error signal {tilde over (y)} by operatingon such error signal with a summing node, feedback network, and gainmultipliers implementing the transfer function N_(dc)(s)/D_(dc)(s). Thelinear controller 14 also generates a transformed signal {tilde over(y)}_(ad) based on the tracking error signal {tilde over (y)} byoperating on such error signal with a summing node, one or moreintegrators, and a feedback network from output terminal(s) of theintegrator(s) that have gain multiplier(s) implementing the transferfunction N_(ad)(s)/D_(dc)(s). The linear controller 14 generates thetransformed signal {tilde over (y)}_(ad) so that the transfer functionfrom the transformed signal {tilde over (y)}_(ad) to the adaptivecontrol signal v_(ad) is strictly positive real (SPR). The linearcontroller 14 is coupled to supply the transformed signal {tilde over(y)}_(ad) to the AE 16.

More specifically, the error conditioning element 38 of the AE 16 iscoupled to receive the transformed signal {tilde over (y)}_(ad). Theerror conditioning element 38 is also coupled to receive basisfunction(s) φ and generates the training signal δ based on the signal{tilde over (y)}_(ad) and the basis function(s). The error conditioningelement 38 can generate the training signal δ by filtering the basisfunction(s) φ and multiplying the resulting signal φ_(f) by thetransformed signal {tilde over (y)}_(ad). The error conditioning element38 is coupled to supply the training signal δ to the neural networkadaptive element (NNAE) 36.

The NNAE 36 uses the training signal δ to adjust connection weights W ofits neural network to adapt to plant dynamics that are unmodeled andtherefore not adapted to by the LC 14. The delay element 40 of the AE 16is coupled to receive the pseudo-control signal v and the plant outputsignal y and is coupled to supply non-delayed signals v, y, and delayedversions v_(d), y_(d) thereof generated by the delay element 40 based onthe signals v, y to the NNAE 36. The delayed signals y_(d) are delayedrelative to the plant output signal y by time delay increments d to(n−1) d, n being the number of the full plant states, although inpractice fewer or more such delays may be used. The delayed signal(s)v_(d) are delayed relative to the pseudo-control signal v by time delayincrements d to (n−r−1) d, r being the relative degree. The use of thesedelayed signals assures that the weight coefficient(s), W, and errorsignal(s) {tilde over (y)} are bounded so that the ACS 10 is stable.

Based on the connection weight(s) W determined by training signal δ, andthe pseudo-control signal v and delayed version(s) v_(d) thereof and/orthe plant output signal y and delayed version(s) y_(d) thereof suppliedas inputs to the NNAE 36, the NNAE generates the adaptive control signalv_(ad). The NNAE 36 is coupled to supply the adaptive control signalv_(ad) to the summing unit 22. The summing unit 22 is also coupled toreceive the pseudo-control component signal v_(dc) from the linearcontroller 14 and the r-th time derivative of the commanded state signaly_(c) ^((r)) from the command filter unit 28. Based on the signals y_(c)^((r)), v_(dc), v_(ad), the summing unit 22 generates the pseudo-controlsignal v. The summing unit 22 is coupled to supply the pseudo-controlsignal v to the model inversion unit 18.

The model inversion unit 18 is also coupled to receive the plant outputsignal y. The model inversion unit 18 generates a command control signalδ_(c) based on the pseudo-control signal v and the plant output signaly. More specifically, the model inversion unit 18 subjects the signalsv, y to a function that inverts the linear control model implemented bythe linear controller 14, to produce the command control signal δ_(c).The model inversion unit 18 is coupled to supply the resulting commandcontrol signal δ_(c) to the actuator(s) 32 of the plant 12. Theactuator(s) 32 are coupled to or associated with the controllednonlinear system 30 so that the control the state(s) of such system,based on the command control signal δ_(c). The sensor(s) 34 are coupledor associated with the controlled nonlinear system 30, and generates theplant output signal y that is a function of the full plant state(s) ofthe controlled nonlinear system 30.

Model inversion in the unit 18 can be performed in the following way.For the scalar case (p=1) if the regulated output, y, has relativedegree r, then the output equation can be differentiated r times withrespect to time until the control appears explicitly. In this case, weassume that we have a model of this r^(th) derivativey ^((r)) =ĥ _(r)(y,δ _(c))=v  (5)

Thus the model inversion of the unit 18 is defined byδ_(c) =ĥ _(r) ⁻¹(y,v).  (6)

2. Specific Description of Adaptive Control System and Method

Let the dynamics of an observable nonlinear single-input-single-output(SISO) system be given by the following equations:{umlaut over (x)}=f(x,δ _(c)), y=h(x)  (7)

-   -   where xεΩ⊂        ^(n) is the state of the system, δ_(c), yε        the system input (control) and output (measurement) signals,        respectively, and f(·,·), h(·)εC∞ are unknown functions.        Moreover, n need not be known.

To ensure proper operation, the following assumption is made in theimplementation of the ACS 10.

Assumption 1. The dynamical system of Eq. (7) satisfies the outputfeedback linearization conditions Isidori, A. (1995), Nonlinear ControlSystems, Springer-Verlag, Inc. with relative degree r, i.e.,$\begin{matrix}{{{y^{(r)} = {{h_{r}( {x,\delta_{c}} )}.{Here}}},{h_{r}\underset{=}{\Delta}\frac{\mathbb{d}^{r}h}{\mathbb{d}t^{r}}},{{{such}\quad{that}\quad\frac{\partial h_{i}}{\partial u}} = {{0\quad{for}\quad 0} \leq i < {r\quad{and}}}}}{\frac{\partial h_{r}}{\partial u} \neq 0.}} & (8)\end{matrix}$

This disclosure addresses the design of an output feedback control lawthat utilizes the available measurement y(t), to obtain system outputtracking of a bounded trajectory y_(c)(t) that is assumed to be r-timesdifferentiable, i.e., y_(c)εC^(r). The difference between unknowndynamics function h_(r) and its estimate ĥ_(r), or the modeling error,is mapped using a NN. This mapping has to be based on measured input andoutput data only. To this end, the universal approximation property ofneural networks and the observability of the system are utilized toconstruct this mapping on-line using measured input/output timehistories. These various features of the proposed control design schemeare presented in the next section.

3. Controller Design 3.1 Feedback Linearization

Feedback linearization is performed by introducing the transformationv=ĥ _(r)(y,δ _(c)),  (9)where v is commonly referred to as a pseudo control signal, andĥ_(r)(y,δ_(c)) is the best available approximation of h_(r)(y,δ_(c)).Then, the system dynamics can be expressed asy ^((r)) =v+Δ′,  (10)whereΔ′=Δ′(x,δ _(c))=h _(r)(x,δ _(c))−ĥ _(r)(y,δ _(c)).  (11)Using this transformation, Eq. (10) represents the dynamic relation of rintegrators between the pseudo-control signal v and the plant outputsignal y, where the error Δ′ acts as a disturbance signal. Assuming thatthe plant output signal y is required to track a known bounded inputcommand signal y_(c), the pseudo-control signal v is chosen to have theformv=y _(c) ^((r)) +v _(dc) −v _(ad),  (12)where v_(dc) is the output of a stabilizing linear dynamic compensatorfor the linearized dynamics in Eq. (10) with Δ′=0, and v_(ad) is theadaptive control signal designed to approximately cancel Δ′. The r-thderivative of the input signal, y_(c) ^((r)), is introduced as afeedforward term to condition the error dynamics. This derivative can beeasily obtained if the tracking (or command) signal y_(c) is generatedusing an r-th (or higher) order reference model forced by an externalinput. The reference model serves to define the desired response of theclosed loop system. The input to the dynamic compensator is the trackingerror, which is defined by{tilde over (y)}=y _(c) −y.  (13)It is important to point out that the model approximation functionĥ_(r)(·,·) should be defined so that it is invertible with respect to u,allowing the actual control input to be computed byδ_(c) =ĥ _(r) ⁻¹(y,v).  (14)Clearly, the accuracy of the approximation h_(r)(x,ĥ_(r) ⁻¹(y,v))≈v isgoverned byΔ′(x,δ _(c))=Δ(x,y,v)=h _(r)(x,ĥ _(r) ⁻¹(y,v))−v.  (15)From Eq. (11) and Eq. (15), notice that Δ depends on v_(ad) through v,whereas v_(ad) has to be designed to cancel Δ. The following assumptionis introduced to guarantee existence and uniqueness of a solution forv_(ad):

Assumption 2. The map v_(ad)|→Δ is a contraction over the entire inputdomain of interest.

Using Eq. (11), the condition in Assumption 2 implies: $\begin{matrix}{{\frac{\partial\Delta}{\partial v_{ad}}} = {{{\frac{{{\partial\text{(}}h_{r}} - {{\hat{h}}_{r}\text{)}}}{\partial u}\frac{\partial\delta_{c}}{\partial v}\frac{\partial v}{\partial v_{ad}}}} = {{{\frac{{{\partial\text{(}}h_{r}} - {{\hat{h}}_{r}\text{)}}}{\partial u}\frac{\partial\delta_{c}}{\partial h_{r}}}} < 1}}} & (16)\end{matrix}$which can be re-written in the following way: $\begin{matrix}{{\frac{{\partial h_{r}}/{\partial\delta_{c}}}{{\partial{\hat{h}}_{r}}/{\partial\delta_{c}}}} < 1} & (17)\end{matrix}$The condition (17) is equivalent to the following two conditions(1) sgn(∂h _(r) /∂δ _(c))=sgn(∂ĥ _(r)/∂δ_(c))(2) |∂ĥ _(r)/∂δ_(c) |>|∂h _(r)/∂δ_(c)|/2>0.The first condition means that control reversal is not permitted, andthe second condition places a lower bound on the estimate of the controleffectiveness in (14).

3.2 Control System Architecture

Based on the above description, the overall control system architectureis presented in FIG. 1. The central components of the system are: (a)the model inversion/linearization unit 18 implementing block ĥ_(r)⁻¹(y,δ_(c)), (b) the adaptive neural network based element 16 isdesigned to minimize the effect of Δ, and (c) the linear dynamiccompensator of the linear controller 14. The input into the ACS 10 isthe reference command tracking signal y_(c) and its r-th derivativey_(c) ^((r)), generated by, e.g., a reference model forced by anexternal input.

It is important to note the two output signals (v_(dc), {tilde over(y)}_(ad)) of the linear compensator. The pseudo-control componentsignal, v_(dc), is designed to stabilize the linearized system, asdescribed earlier. The transformed signal, {tilde over (y)}_(ad), is alinear combination of the compensator states and its input, i.e., thetracking error signal {tilde over (y)}. This signal is generated toensure an implementable error signal δ that is used to adapt the NNweights W of the NNAE 36.

3.3 Tracking Error Signal Analysis

The analysis presented in this subsection is carried out to facilitatethe design of the NNAE 36 and the second output signal {tilde over(y)}_(ad) of the linear dynamic compensator 14. To formulate the overalltracking error dynamics of the controlled system, the specific choice ofthe pseudo-control signal v is given by Eq. (12) is substituted into Eq.(16), leading toy ^((r)) =y _(c) ^((r)) +v _(dc) −v _(ad)+Δ,  (18)or alternately{tilde over (y)} ^((r)) =−v _(dc) +v _(ad)−Δ.  (19)

These error dynamics are depicted schematically in FIG. 2. Morespecifically, under tracking error dynamics analysis, the ACS 10 reducesto a summing node 42, an integrator 44, and a linear dynamic compensator(LDC) 46. The summing node 42 is coupled to receive the signal v_(ad)−Δand the pseudo-control component signal v_(dc). The summing node 42subtracts the signal v_(dc) from the signal v_(ad)−Δ to generate thesignal {tilde over (y)}^((r)). The r-th degree integrator 44 integratesthe signal {tilde over (y)}^((r)) to produce the signal {tilde over(y)}. The LDC 46 is coupled to receive the signal {tilde over (y)} fromthe integrator 44. Based on the signal {tilde over (y)}, the LDC 46generates the signal v_(dc) that is fedback to the summing node 42. TheLDC 46 also generates the signal {tilde over (y)}_(ad) based on thesignal {tilde over (y)}. The LDC 46 implements a transfer function thatis SPR to map the signal {tilde over (y)} to the signal v_(ad)−Δ toensure stability of the ADC 10.

The single-input two-output transfer matrix of the linear dynamiccompensator is denoted by $\begin{matrix}{\begin{Bmatrix}{v_{d\quad c}(s)} \\{{\overset{\sim}{y}}_{ad}(s)}\end{Bmatrix} = {{\frac{1}{D_{d\quad c}(s)}\begin{bmatrix}{N_{dc}(s)} \\{N_{ad}(s)}\end{bmatrix}}{\overset{\sim}{y}(s)}}} & (20)\end{matrix}$where s represents the complex Laplace variable. The LDC 46 can comprisetransfer function elements 48, 50. The transfer function element 48 canbe used to implement the transfer function N_(dc)(s)/D_(dc)(s) mappingthe signal y to the signal v_(dc). The transfer function element 50 canbe used to implement the transfer function N_(ad)(s)/D_(dc)(s) to mapthe signal {tilde over (y)} to the signal {tilde over (y)}_(ad). Furtherdetails regarding the LDC 46 are described below.

Assumption 3. The linearized system in FIG. 2 is stabilized using astable linear dynamic compensator 46, i.e., the roots of the denominatorpolynomial D_(dc)(s) are located in the open left halfplane of thecomplex plane s.

Since the linearized system dynamics, and hence the error dynamics,consist of r pure integrators, this assumption introduces only a verymild restriction on the design. Based on the compensator defined in Eq.(20), the closed loop transfer function of the system depicted in FIG. 2is given by: $\begin{matrix}{{{\overset{\sim}{y}}_{ad}(s)} = {\frac{N_{ad}(s)}{{s^{r}{D_{dc}(s)}} + {N_{dc}(s)}}( {v_{ad} - \Delta} )(s)\underset{=}{\Delta}{G(s)}( {v_{ad} - \Delta} ){(s).}}} & (21)\end{matrix}$Analyzing the denominator of Eq. (21), the Routh-Hurwitz stabilitycriterion implies that a necessary condition for closed loop systemstability is that the degree of the compensator numerator, N_(dc)(s),and hence of its denominator, D_(dc)(s), should be at least (r−1), i.e.,qΔ deg(D _(dc)(s))≧deg(N _(dc)(s))≧r−1.  (22)This dictates the design of the linear dynamic compensator$\begin{matrix}{{{v_{dc}(s)} = {\frac{N_{dc}(s)}{D_{dc}(s)}{\overset{\sim}{y}(s)}}},} & (23)\end{matrix}$which can be carried out using any linear control design technique(classical, pole placement, optimal LQ, etc.), with the constraint ofassumption 3. The numerator N_(ad)(s), associated with the output {tildeover (y)}_(ad), does not affect the stability of the error system ofFIG. 2.

3.4 Neural Network Based Approximation

The term “artificial neural network” has come to mean any architecturethat has massively parallel interconnections of simple “neural”processors. Given xε

^(N) ¹ , a three layer-layer NN has an output given by: $\begin{matrix}{{y_{i} = {\sum\limits_{j = 1}^{N_{2}}\lbrack {{w_{ij}{\phi( {{\sum\limits_{k = 1}^{N_{1}}{v_{jk}x_{k}}} + \theta_{vj}} )}} + \theta_{wi}} \rbrack}},{i = 1},\ldots\quad,N_{3}} & (24)\end{matrix}$where φ(·) is the activation function, v_(jk) are the first-to-secondlayer interconnection weights, and _(wi) are the second-to-third layerinterconnection weights. θ_(vj) and θ_(wi) are bias terms. Such anarchitecture is known to be a universal approximator of continuousnonlinearities with squashing activation functions. See Funahashi, K.(1989), On the approximate realization of continuous mappings by neuralnetworks. Neural Networks, 2:183-192; Homik, K., Stinchcombe, M., andWhite, H. (1989), Multilayer feedforward networks are universalapproximators, Neural Networks, 2:359-366.

Linearly parameterized neural networksy=W ^(T)φ(x)  (25)are universal approximators as well, if vector function φ(·) can beselected as a basis over the domain of approximation. Then a generalfunction f(x)εC^(k), xεD⊂

^(n) can be written asf(x)=W ^(T)φ(x)+ε(x),  (26)where ε(x) is the functional reconstruction error. In general, given aconstant real number ε*>0, f(x) is within ε* range of the NN, if thereexist constant weights W, such that for all xε

^(n). Eq. (20) holds with ∥ε∥<ε*.

Definition 1. The functional range of NNAE 36 is dense over a compactdomain xεD, if for any f(·)εC^(k) and ε* there exists a finite set ofbounded weights W, such that Eq. (26) holds with ∥ε∥<ε*.

Various publications show that the functional range of NN in Eq. (25) isdense for different activation functions φ(·). See Cybenko, G. (1989)publication. Approximation by superpositions of sigmoidal function,Mathematics of Control, Signals, Systems, 2(4):303-314; Park, J. andSandberg, I. (1991), Universal approximation using radial basis functionnetworks, Neural Computation, 3:246-257; Sanner, R. and Slotine, J.(1992), Gaussian networks for direct adaptive control, IEEE Transactionson Neural Networks, 3(6):837-864.

The following theorem extends these results to map the unknown dynamicsof an observable plant from available input/output history.

Theorem 1. Given ε*>0, there exists a set of bounded weights W, suchthat Δ(x,y,v), associated with the system (1)-(5), can be approximatedover a compact domain D⊂Ω×R by a linearly parameterized neural networkΔ=W ^(T)φ(η)+ε(η),∥ε∥<ε*(η),  (27)****using the input vectorη(t)=[1{overscore (v)} _(d) ^(T)(t){overscore (y)} _(d)^(T)(t)]^(T),  (28)where{overscore (v)} _(d) ^(T)(t)=[v(t)v(t−d) . . . v(t−(n ₁ −r−1)d)]^(T){overscore (y)} _(d) ^(T)(t)=[y(t)y(t−d) . . . y(t−(n ₁−1)d)]^(T)with n₁≧n and d>0, provided there exists a suitable basis of activationfunctions φ(·) on the compact domain D.

The output of the adaptive element 16 in FIG. 1 is designed asv _(ad)=Ŵ^(T)φ(η),  (29)where W are the estimates of the weights. Eq. (29) will always have atleast one fixed-point solution, so long as φ(·) is made up of boundedbasis functions.

3.5 Construction of SPR Transfer Function

As discussed earlier, the second output of the linear dynamiccompensator 46, {tilde over (y)}_(ad), will be used to construct therule for adapting Ŵ in Eq. (29). Using Eqs. (27) and (29) in Eq. (21)implies:{tilde over (y)} _(ad)(s)=G(s)({tilde over (W)} ^(T)φ(η)−ε)  (30)where {tilde over (W)}=Ŵ−W is the weight error. As will be seen in thenext section, for the NN adaptation rule to be realizable, i.e.dependent on available data only, the transfer function G(s) must bestrictly positive real (SPR). However, the relative degree of G(s) is atleast r. When the relative degree of G(s) is one, it can be made SPR bya proper construction of N_(ad)(s). If r>1, G(s) cannot be SPR throughthis technique alone.

To achieve SPR in the r>1 case, following the Kim and Lewis, 1998publication, a stable low pass filter T⁻¹(s) is introduced in Eq. (30)as:{tilde over (y)} _(ad)(s)=G(s)T(s)({tilde over (W)}^(T)φ_(f)+δ−ε_(f))(s)  (31)where φ_(f) and ε_(f) are the signals φ and ε, respectively, after beingfiltered through T⁻¹(s), and δ_(m)(s) is the “so-called” mismatch termgiven byδ_(m)(s)=T ⁻¹(s)({tilde over (W)} ^(T)φ)−{tilde over (W)}^(T)φ_(f)  (32)that can be bounded as∥δ_(m)(t)∥≦c∥{tilde over (W)}∥ _(F), c>0.  (33)

The numerator of the transfer function G(s)T(s)={overscore (G)}(s) inEq. (31) is T(s)N_(ad)(s). The polynomial T(s) is Hurwitz, but otherwisecan be freely chosen, along with the numerator polynomial N_(ad)(s) ofEq. (21) that defines the output {tilde over (y)}_(ad). Hence, thenumerator polynomial (or the zeros) of {overscore (G)}(s) can be freelychosen to make it SPR. Two approaches can be utilized in constructingT(s) and N_(ad)(s) to make {overscore (G)}(s) SPR. To avoid anunrealizable feedthrough, {overscore (G)}(s) will be assigned (r+q−1)zeros, thus making it relative degree one.

Zero placement approach: Since {overscore (G)}(s) is a stable transferfunction, its zeros can be easily placed to make it SPR, e.g., byinterlacing them with its poles. From Bode plot analysis it is easy toconclude that such a pole-zero pattern will ensure a phase shift in therange of ±90°.

LKY approach: Assume that $\begin{matrix}{{\overset{\_}{G}(s)} = \frac{{b_{1}s^{p - 1}} + {b_{2}s^{p - 2}} + \ldots + b_{p}}{s^{p} + {a_{1}s^{p - 1}} + \ldots + a_{p}}} & (34)\end{matrix}$where p=r+q is the number of the closed loop poles. The controllercanonical state space realization of this transfer function is given by$\begin{matrix}{{\overset{.}{z} = {{A_{c1}z} + {B_{c1}( {{{\overset{\sim}{W}}^{T}\phi_{f}} + \delta - ɛ_{f}} )}}}{{{\overset{\sim}{y}}_{ad} = {C_{c1}z}},{where}}{A_{c1} = {{\begin{bmatrix}{- a_{1}} & {- a_{2}} & \cdots & \quad & \quad & {- a_{p}} \\1 & 0 & \cdots & \quad & \quad & 0 \\0 & ⋰ & ⋰ & \quad & \quad & \vdots \\\vdots & ⋰ & ⋰ & ⋰ & \quad & \vdots \\0 & \cdots & 0 & 1 & \quad & 0\end{bmatrix}\quad B_{c1}} = \begin{bmatrix}1 \\0 \\0 \\\vdots \\0\end{bmatrix}}}{C_{c1} = \lfloor \begin{matrix}b_{1} & b_{2} & \cdots & b_{p}\end{matrix} \rfloor}} & (35)\end{matrix}${overscore (G)}(s) is SPR if and only if it complies with theLefschetz-Kalman-Yakubovitz (LKY) Lemma, Ioannou, P. A. and Sun, J.(1996), Robust Adaptive Control, Prentice Hall, Inc., p. 129, i.e.,there exists Q>0 such that the solution P ofA _(c1) ^(T) P+PA _(c1) =−Q  (36)is positive definite andPB _(c1) =C _(c1) ^(T).  (37){overscore (G)}(s) can be constructed utilizing the LKY condition asfollows:

-   -   a) Choose Q>0 and solve Eq. (36) for P>0.    -   b) Using Eq. (37), compute C_(c1), which in this canonical form        is simply the first column of P. From Eqs. (33) and (34), the        elements of C_(c1) are also the coefficients of the numerator        polynomial of {overscore (G)}(s). Since {overscore (G)}(s) is        SPR, it is guaranteed that this numerator is Hurwitz.    -   c) Solve the numerator polynomial for its roots.        From the zeros obtained by either of the above methods, choose        (r−1) of these to construct T(s), while the remaining q zeros        makeup N_(ad)(s). The fact that the numerator of {overscore        (G)}(s) is Hurwitz ensures also that T(s) and N_(ad)(s) are        individually Hurwitz. There is freedom in scaling T(s) and        N_(ad)((s), which could be utilized to normalize the maximum        gain of T⁻¹(s).

To summarize, N_(dc)(s)/D_(dc)(s) is designed to stabilize thelinearized system dynamics, while N_(ad)(s) is constructed to meet theSPR condition needed for a realizable implementation.

Neural Network Adaptation Rule

As is evident from Eq. (31), the filter T⁻¹(s) should operate on all thecomponents of the NN vector φ. All these filters can be cast in onestate space realization:{dot over (z)} _(f) =A _(f) z _(f) +B _(f)φφ_(f) =C _(f) z _(f),  (38)where the diagonal blocks of the state space matrices (A_(f), B_(f),C_(f)) are constructed from a state space realization of the filterT⁻¹(s). Since the filter is stable, ∃P_(f)>0, satisfyingA _(f) ^(T) P _(f) +P _(f) A _(f) =−Q _(f)  (39)for any positive definite Q_(f)>0.The signals φ_(f) are used in the following NN W weight adaptation ruledŴ/dt=−F└{tilde over (y)} _(ad)φ_(f)+λ_(w) Ŵ┘,  (40)where F>0 and λ_(w>)0 are the adaptation gains. In the next section itis proven that this adaptation rule ensures boundedness of the systemerror signals and the network weights. The NNAE 16 of FIG. 1 is depictedin more detail in FIG. 4.

FIG. 4 is an exemplary embodiment of the adaptive element 16 provided byway of example and not limitation as to possible implementations of theNNAE 16. The time delay element 40 comprises one or more time-delay(TDL) units 52 ₁-52 _(N1-r-1) coupled to receive the pseudo-controlsignal v, and TDL units 54 ₁-54 _(N1-1) coupled to receive the plantoutput signal y. The TDL units 52 ₁-52 _(N1-r-1), 54 ₁-54 _(N1-1)generate delayed versions v_(d), y_(d) of the signals v, y, and arecoupled to supply these delayed signals v_(d), y_(d) as well asundelayed signals v, y, to the NNAE 36. The neural network (NN) 64 ofthe NNAE 36 multiplies the signals v, v_(d), y, y_(d) by respectiveweight data V and transmits the resulting signals to respective basisfunctions φ(·) 56 ₁, 56 ₂, . . . , 56 _(N1). The basis functions φ(·) 56₁, 56 ₂, . . . , 56 _(N1) are coupled to receive V-weighted signals v,v_(d), y, y_(d) and generate respective signals based thereon. Thegenerated signals are multiplied by respective weight data W and summedat respective summation nodes 58 ₁, 58 ₂, . . . , 58 _(N2). The NNAE 36is coupled to supply the resulting summed signals as the vector signalv_(ad) to the summing node 22 of FIG. 1 for generation of thepseudo-control signal v.

To ensure boundedness of the basis functions φ(·) 56 ₁, 56 ₂, . . . , 56_(N1) and neural network weights V, W, the NNAE 36 is coupled to supplythe basis functions φ(·) 56 ₁, 56 ₂, . . . , 56 _(N1) as signals to theerror conditioning element 38. The error conditioning element 38comprises a filter 60 and a multiplier 62. The filter 60 operates on thebasis functions 56 ₁, 56 ₂, . . . , 56 _(N1) with a filtering transferfunction T⁻¹(s) as previously described with respect to Equation (32) togenerate filtered basis functions φ_(f)(·). The filter 60 is coupled tosupply the filtered basis functions φ_(f)(·) to the multiplier 62. Themultiplier 62 is also coupled to receive the transformed signal y_(ad).The multiplier 62 generates the signal δ that is a vector product of thesignals φ_(f)(·), y_(ad). The multiplier 62 is coupled to supply thesignal δ to the NNAE 36. Based on the signal δ, the NNAE 36 adjusts theweight data W to adapt the NNAE 36 to generate the pseudo-control signalso as to compensate for error Δ associated with the command controlsignal δ_(c).

4. Boundedness Statement

The following theorem establishes sufficient conditions for boundednessof the error signals and neural network weights in the proposedclosed-loop adaptive output feedback architecture.

Theorem 2. Subject to assumptions 1-3, the error signals of the systemcomprised of the dynamics in Eq. (7), together with the dynamicsassociated with the realization of the controller in Eq. (14) and the NNadaptation rule in Eq. (40), are uniformly ultimately bounded, providedthe following conditions holdQ _(m)>2∥C _(c1)∥, λ_(W) >c ²/4,  (41)where Q_(m) is the minimum eigenvalue of Q.

5. Processor-Based Embodiment of Adaptive Control System Using DirectOutput Feedback

Although it is possible to implement the elements 14, 16, 18, 20, and 22of the ACS 10 of FIG. 1 as discrete or grouped analog or digitaldevices, these elements can alternatively be implemented in aprocessor-based ACS system 10. The processor-based system 10 includes aprocessor 66 and memory 69 storing data and a control program, toimplement the elements 14, 16, 18, 20, and 22. More specifically, thecontrol program can be implemented as software objects or modules thatperform the functions of the elements 14, 16, 18, 20, 22 as previouslydescribed. The data can be parameters such as the NN connection weightsW, V and/or basis function(s) φ that are updated by the processor 66, aswell as temporary data and intermediate calculations, commanded statesignal levels, plant output signal levels, etc. The ACS 10 of FIG. 5 canfurther comprise bus 70 to which the operator interface unit 26, thecommand filter unit 28, the actuator 32, the sensor(s) 34, the processor66, and the memory 68, are coupled.

In operation, the sensor(s) 34 generate plant output signal y and supplythis signal to the operator interface unit 26 via the bus 70. Theoperation interface unit 26 generates a signal readable or discernibleby the operator. If the operator 24 is human, the operator interfaceunit 26 can generate a display or the like based on the plant outputstate signal y. If the operator 24 is a processor or machine, theoperator interface unit 26 can convert the plant output state signal yinto a format usable by the operator. The operator 24 if human producesone or more signals through control actions applied to a command filterunit 28. For example, in the case of an aircraft, the control actionsmay be applied to control instruments of the aircraft. Alternatively, ifthe operator 24 is a machine, the command signal produced by theoperator can be applied to the command filter unit 28. The commandfilter unit 28 generates the commanded output signal y_(c) and the rthderivative of the commanded output signal y_(c) ^((r)). The commandfilter unit 28 supplies the signals y_(c), y_(c) ^((r)) to the processor66 or to the memory 68 at a location accessible to the processor 66. Thesensor(s) 34 can supply the plant output signal y directly to theprocessor 66, or to the memory 68 at a location accessible to theprocessor 66 via the bus 70. The processor 66 performs the functions ofthe elements 14, 16, 18, 20, 22 to generate a command control signalδ_(c). The processor 66 is coupled to supply the command control signalδ_(c) to the actuator(s) 32 via the bus 70. The actuator(s) 32 performcontrol of the plant 12 in a manner that can affect the plant state(s).The sensor(s) 34 sense and generate the plant output signal y for thenext control cycle. Processing performed by the processor 66 inexecuting its control program can be repeated over successive controlcycles as long as required to control the plant 12.

6. General Method of the Invention

FIG. 6 is a flowchart of processing performed by the ACS 10 of FIGS.1-5. In step S1 of FIG. 6 the method begins. In step S2 a commandcontrol signal δ_(c) is generated by inverting an approximate model ofthe plant dynamics, based on a pseudo-control signal v and the plantoutput signal y. In step S3 the command control signal δ_(c) is suppliedto control the plant. In step S4 the plant output signal y is generatedby the sensors. In step S5 a tracking error signal {tilde over (y)} isgenerated by differencing corresponding signal components of thecommanded output signal y_(c) and optional derivative(s) thereof, andthe plant output signal y. In step S6 a pseudo-control component signalv_(dc) is generated based on the tracking error signal {tilde over (y)}using the transfer function N_(dc)(s)/D_(dc)(s). In step S7 atransformed signal {tilde over (y)}_(ad) is generated based on thetracking error signal {tilde over (y)} using transfer functionN_(ad)(s)/D_(dc)(s). In step S8 the rth derivative of the commandedoutput signal y_(c) ^((r)) is generated. In step S9 the signal {tildeover (y)}_(ad) is generated to render the transfer function from thesignal v_(ad) to the signal {tilde over (y)}_(ad) strictly positive realby appropriate choice of N_(ad)(s). In step S10 a training signal δ isgenerated by filtering basis function(s) φ and multiplying the filteredbasis function(s) φ by the transformed signal y_(ad). In step S11connection weights W of a neural network are updated in a bounded mannerusing the training signal δ. In step S12 delayed versions of thepseudo-control signal v are generated. In step S13 delayed versions ofthe plant output signal y are generated. In step S14 the adaptivecontrol signal v_(ad) is generated based on the pseudo-control signal v,delayed versions v_(d) of the signal v, plant output signal y, plantoutput signal y_(d), connection weights W, V, and basis function(s) φupdated based on the training error signal δ. In step S15 apseudo-control signal v is generated based on the rth time-derivative ofthe commanded output signal y_(c) ^((r)) pseudo-control component signalv_(dc), and adaptive control signal v_(ad). In step S16 the method ofFIG. 6 ends.

7. Example of Implementation of the Adaptive Control System HavingDirect Output Feedback Control

The performance of the ACS 10 using output feedback is demonstrated byconsidering the following nonlinear system, consisting of a modified Vander Pol oscillator coupled to a lightly damped mode{dot over (x)}₁=x₂  (42){dot over (x)} ₂=−2(x ₁ ²−1)x ₂ −x ₁ +u  (43){dot over (x)}₃=x₄  (44){dot over (x)} ₄ =−x ₃−0.2x ₄ +x ₁  (45)y=x ₁ +x ₃  (46)

The output y has a relative degree of r=2. From a practical perspective,the system can be thought of as a second order nonlinear plant model,whose realization consists of states x₁ and x₂, in which the output ismodeled as y=x₁. However, the system contains also a very lightly dampedunmodeled mode, with a natural frequency equal to that of the linearizedplant. This mode is excited by the plant displacement state (x₁) and iscoupled to the measurement.

The output signal y does not have a full relative degree in the presenceof the unmodeled mode. The low natural frequency of this mode isencompassed by the bandwidth of the controlled system. This introduces achallenging control problem, in particular for methods that require theoutput to have a full relative degree. Moreover, this example is treatedas if even the Van der Pol model is unknown, and only the fact that r=2is given. This is not an unreasonable assumption in that in manysystems, the number of plant states and hence the value of r can bededuced from knowledge of the behavior of the plant. Thus, thecontroller design is performed assuming ÿ=u, implying that in FIG. 1 theplant transfer function from the pseudo-control signal v toy is 1/s².

A first order lead-lag compensator was selected to stabilize theassociated error dynamics. In addition, the first design approachdescribed in Section 3.5 was used to satisfy the SPR condition. Theresulting two outputs of the compensator are given by $\begin{matrix}{\begin{Bmatrix}{v_{d\quad c}(s)} \\{{\overset{\sim}{y}}_{ad}(s)}\end{Bmatrix} = {{\frac{1}{s + 5}\begin{bmatrix}{8( {s + 0.75} )} \\{20( {s + 1} )}\end{bmatrix}}{\overset{\sim}{y}(s)}}} & (50)\end{matrix}$which places the closed loop poles of the error dynamics at −3, −1±j.The low pass filter 60 T⁻¹(s) discussed in Eq. (26) was chosen as$\begin{matrix}{{T^{- 1}(s)} = {\frac{1}{{0.5s} + 1}.}} & (51)\end{matrix}$It is easy to verify that the transfer function G(s)T(s) of Eq. (31) isSPR.

A Gaussian Radial Basis Function (RBF) NN with only three neurons and abias term was used in the adaptive element. The functional form for eachRBF neuron was defined byφ_(i)(η)=e ⁻(η−η_(ci))^(T)(n−n _(ci))/σ², σ={square root}{square rootover (2)}, i=1, 2, 3.  (52)The centers η_(ci), i=1, 2, 3 were randomly selected over a grid ofpossible values for the vector η. All of the NN inputs were normalizedusing an estimate for their maximum values. The current and two delayedvalues for the plant output signal y and only the current pseudo-controlsignal v were used in the input vector to the neurons. The completeinput vector consisted of these values together with a bias term, asillustrated in FIG. 4. Thus, there are a total of four NN weights in Eq.(29). The network gains were F=50 and λ_(w)=1.

In the simulation, the initial states of the system were set tox₁(0)=0.5, x₂(0)=2.5, x₃(0)=x₄(0)=0. The system was commanded to followthe output of a second order reference model for the MIU 18, designedwith a natural frequency of ω_(n)=1{square root}{square root over (2)}rad/sec and damping ζ=2/{square root}{square root over (2)}, and drivenby a square wave input command signal y_(c).

First, the controlled system performance is evaluated without theunmodeled mode dynamics, i.e., removing Eqs. (44) and (45) and settingthe output y=x₁. However, it will be recalled that the controller hasbeen designed given only the fact that r=2. FIG. 7 compares the systemresponse without NN augmentation (solid line) with the reference modeloutput (dashed line), clearly demonstrating the almost unstableoscillatory behavior caused by the nonlinear elements in the Van del Polequation. FIGS. 8A and 8B show that with NN augmentation, theseoscillations are eliminated after a period of about three seconds. Thisis accounted for by the successful identification of the model inversionerror by the NN, which is also illustrated in FIG. 8B by comparing theNN output (solid line) with the computed inversion error (dashed line).

Next, the effect of the unmodelled dynamics is examined. In this case,the response without the NN is unstable, and therefore is not shown. Theresponse with NN augmentation is presented in FIG. 9. It shows onlyminor performance degradation compared to the full relative degree caseof FIG. 8A. Since the unmodeled mode is well within the bandwidth of thecontrol system (when viewed with, v_(ad)=Δ), this demonstrates that theadaptive system learns to interact with the added mode to achieve goodtracking performance.

In FIGS. 8A, 8B, 9 the NN based adaptive controller exhibits a steadystate tracking error. This error can be removed by introducing anadditional integral control action when designing the linearcompensator. The performance of the controller with integral action isshown in FIGS. 10A and 10B. The steady state tracking error is zero,while the transient response behavior is only slightly compromised. Thebounded NN weight time histories are also depicted in this figure,showing that most of the weight adaptation takes place when the commandreverses direction.

8. Additional Considerations

The stability results are semiglobal in the sense that they are localwith respect to the domain D. If the NN universally approximates theinversion error over the whole space

^(n+1), then these results become global.

The NN update laws consist of a modified gradient algorithm along withthe standard σ-modification term as described in the Kim and Lewis 1998publication. These laws have been proven to be passive in Lewis, F.(1999), Nonlinear network structures for feedback control, Asian Journalof Control, 1(4):205-228.

The NN learning takes place on-line, and no off-line training isrequired. No assumption on persistent excitation is required.

The ultimate bound for the tracking error can be made smaller byincreasing the linear design gains. This will result in increasedinteraction with unknown or unmodeled plant dynamics. However, Theorem 2remains valid so long as assumptions 2 and 3 hold.

In the case of plants of unknown dimension but with known relativedegree, the described methodology applies with a slight modification ofthe input vector to the network: a large range of input/output datashould be used, i.e., n₁>>n.

9. CONCLUSION

The described ACS 10 presents an adaptive output feedback control designprocedure for nonlinear systems, that avoids state estimation. The mainadvantage is that the stability analysis permits systems of arbitrarybut known relative degree. The full dimension of the plant and itsinternal dynamics may be known or poorly modeled. Only mild restrictionsregarding observability and smoothness are imposed. Consequently, theresult is applicable to adaptive control of nonlinear systems withparametric uncertainty and unmodeled dynamics.

Any trademarks listed herein are the property of their respectiveowners, and reference herein to such trademarks is intended only toindicate the source of a particular product or service.

The many features and advantages of the present invention are apparentfrom the detailed specification and it is intended by the appended claimto cover all such features and advantages of the described methods andapparatus which follow in the true scope of the invention. Further,since numerous modifications and changes will readily occur to those ofordinary skill in the art, it is not desired to limit the invention tothe exact implementation and operation illustrated and described.Accordingly, all suitable modifications and equivalents may be resortedto as falling within the scope of the invention.

1. An adaptive control system (ACS) generating at least one controlsignal δ_(c) to control a plant based on a plant output signal y, theACS connected to receive the plant output signal y by feedback from theplant to the ACS, the ACS generating the control signal δ_(c) toregulate the plant output signal y, the plant output signal y being afunction of the full plant state x having known but unrestrictedrelative degree r.
 2. An ACS as claimed in claim 1 wherein the ACScomprises a linear controller contributing to generation of the controlsignal δ_(c) to control the plant based on the plant output signal y andan approximate linear dynamic model, and further comprises an adaptiveelement contributing to generation of the control signal δ_(c) based onthe plant output signal y to control unmodeled plant dynamics usingadaptive control.
 3. An ACS as claimed in claim 2 wherein the adaptiveelement comprises a neural network implementing adaptive control of theplant via the control signal δ_(c) based on the plant output signal y.4. An ACS as claimed in claim 3 wherein the adaptive element uses atleast one time-delayed version y_(d) of the plant output signal y, thatis supplied together with the plant output signal y as inputs to theneural network, the neural network generating an adaptive control signalv_(ad) contributing to generation of the control signal δ_(c) to controlthe plant output y despite unmodeled plant dynamics, based on thetime-delayed signal y_(d) and the plant output signal y, thetime-delayed version signal y_(d) and the plant output signal y, toensure boundedness of the tracking error signal {tilde over (y)}, thetracking error signal {tilde over (y)} being a difference of the plantoutput signal y and a commanded plant output signal y_(c).
 5. An ACS asclaimed in claim 3 wherein the neural network of the adaptive elementcomprises at least one basis function φ and at least one connectionweight W used to generate an adaptive control signal v_(ad) contributingto generation of the command control signal δ_(c), the adaptive elementfurther comprising an error conditioning element coupled to receive thebasis function φ, the error conditioning element filtering the basisfunction φ with a transfer function T⁻¹(s) to produce filtered basisfunction φ_(f) used to modify the connection weight(s) W of the neuralnetwork through feedback to ensure boundedness of the tracking errorsignal {tilde over (y)}.
 6. An ACS as claimed in claim 1 wherein the ACScomprises a command filter unit generating an rth derivative y_(c)^((r)) of the plant output signal y in which r is an integer indicatingthe number of times the plant output signal y must be differentiatedwith respect to time before an explicit dependence on the controlvariable is revealed.
 7. An ACS as claimed in claim 1 wherein the ACScomprises: an error signal generator generating a tracking error signal{tilde over (y)} indicating the difference between the plant outputsignal y and a commanded output signal y_(c); a linear controllercoupled to receive the tracking error signal {tilde over (y)}, thelinear controller generating a transformed signal {tilde over (y)}_(ad)based on the tracking error signal {tilde over (y)}; and an adaptiveelement coupled to receive the transformed signal {tilde over (y)}_(ad)and generating an adaptive control signal v_(ad) based thereon, theadaptive element operating on the transformed signal {tilde over(y)}_(ad) to generate the adaptive signal v_(ad) such that the transferfunction from v_(ad) to {tilde over (y)}_(ad) is strictly positive real(SPR).
 8. An ACS as claimed in claim 1 wherein sensed variablesaffecting the state of the plant, in addition to the plant output signaly, are fed back to the ACS to control the plant.
 9. A linear controllercoupled to receive a tracking error signal {tilde over (y)} that is avector difference of a plant output signal y that is a function of afull plant state having known but unrestricted relative degree r, and acommanded output signal y_(c), the linear controller generating apseudo-control component signal v_(dc) based on a transfer functionN_(dc)(s)/D_(dc)(s) and the tracking error signal {tilde over (y)}, thepseudo-control component signal v_(dc) used by the linear controller tocontrol the plant based on an approximate linear model, and the linearcontroller generating a transformed signal {tilde over (y)}_(ad) basedon a transfer function N_(ad)(s)/D_(dc)(s) and the tracking error signal{tilde over (y)}, the transformed signal {tilde over (y)}_(ad) used foradaptive control of the plant, the transfer functionsN_(dc)(s)/D_(dc)(s) and N_(ad)(s)/D_(dc)(s) selected to assureboundedness of the tracking error signal {tilde over (y)}.
 10. A methodcomprising the step of: a) generating at least one control signal δ_(c)to control a plant based on a plant output signal y, the ACS connectedto receive the plant output signal y by feedback from the plant to theACS, the ACS generating the control signal δ_(c) to regulate the plantoutput signal y, the plant output signal y being a function of the fullplant state x having known but unrestricted relative degree r.
 11. Amethod as claimed in claim 10 wherein the control signal δ_(c) isgenerated in step (a) so as to control the plant output based on anapproximate linear dynamic model, and so as to control the plant despiteunmodeled plant dynamics based on an adaptive control technique.
 12. Amethod as claimed in claim 10 wherein the adaptive control technique isimplemented with a neural network.
 13. A method as claimed in claim 10wherein the command control signal δ_(c) is generated in step (a) basedon sensed variables affecting the state x of the plant in addition tothe plant output signal y.
 14. A method comprising the steps of: a)selecting a transfer function N_(dc)(s)/D_(dc)(s) used in control of aplant based on a plant output signal y that is a function of all statesx existing in the plant, N_(dc)(s) being the numerator and D_(dc)(s)being the denominator of the transfer function N_(dc)(s)/D_(dc)(s)relating the tracking error signal {tilde over (y)} representing avector difference between the plant output signal y and a commandedoutput signal y_(c), to a linear portion of a pseudo-control signalv_(dc) used to control the plant; b) selecting a transfer functionN_(ad)(s)/D_(dc)(s) used in adaptive control of the plant based on theplant output signal y, N_(ad)(s) being the numerator and D_(dc)(s) beingthe denominator of the transfer function N_(ad)(s)/D_(dc)(s) relatingthe tracking error signal y to an adaptive portion of the tracking errorsignal {tilde over (y)}_(ad) used to generate an adaptive portion of thepseudo-control signal v_(ad); said steps (a) and (b) assuringboundedness of the tracking error signal {tilde over (y)}; and c)physically controlling the plant based on the linear portion of thepseudo-control signal v_(dc) and the adaptive portion of thepseudo-control signal v_(ad) based on the selected transfer functionsN_(dc)(s)/D_(dc)(s) and N_(ad)(s)/D_(dc)(s) and the plant output signaly.
 15. A method comprising the steps of: a) generating a tracking errorsignal {tilde over (y)} that is a vector difference of a plant outputsignal y that is a function of all states x existing in a plant, and acommanded output signal y_(c); b) generating a pseudo-control componentsignal v_(dc) based on a transfer function N_(dc)(s)/D_(dc)(s) and thetracking error signal {tilde over (y)}; c) generating a transformedsignal {tilde over (y)}_(ad) based on a transfer functionN_(ad)(s)/D_(dc)(s) and the tracking error signal {tilde over (y)}; d)controlling the plant with the pseudo-control component signal v_(dc),the pseudo-control component signal v_(dc) controlling the plant basedon an approximate linear model; and e) controlling the plant adaptivelybased on the transformed signal {tilde over (y)}_(ad) used for adaptivecontrol of the plant.
 16. A method as claimed in claim 15 furthercomprising the steps of: f) receiving a plant output signal y that is afunction of all states x existing in a plant; g) delaying the plantoutput signal y to produce a delayed signal y_(d); h) receiving apseudo-control signal v used to control the plant; i) delaying thepseudo-control signal v to produce a delayed signal v_(d); and j)supplying the signals Y, y_(d), v, v_(d) to a neural network to generatean adaptive control signal v_(ad) to control the plant.
 17. A method asclaimed in claim 16 further comprising the steps of: k) filtering atleast one basis function φ to generate a filtered basis function φ_(f);l) multiplying the filtered basis function φ_(f) by the transformedsignal {tilde over (y)}_(ad) to produce an error signal δ; and m)modifying at least one connection weight W of the neural network basedon the error signal δ.
 18. A method as claimed in claim 17 furthercomprising the steps of: n) differentiating the plant output signal y rtimes to produce an rth derivative signal y^((r)) _(c) of the plantoutput signal y, r being the relative degree of the plant output signal;o) summing the rth derivative signal, the pseudo-control componentsignal v_(dc), and the adaptive control signal v_(ad), to generate apseudo-control signal v; and p) generating a command control signalδ_(c) based on the pseudo-control signal v and the plant output signal yby model inversion.
 19. A method comprising the steps of: a) receiving aplant output signal y that is a function of all states existing in aplant; b) delaying the plant output signal y to produce a delayed signaly_(d); c) receiving a pseudo-control signal v used to control the plant;d) delaying the pseudo-control signal v to produce a delayed signalv_(d); and e) supplying the signals y, y_(d), v, v_(d) to a neuralnetwork to generate an adaptive control signal v_(ad) to assist a linearcontroller in controlling the plant.
 20. A method as claimed in claim 19further comprising the steps of: f) filtering at least one basisfunction φ to generated a filtered basis function φ_(f); g) multiplyingthe filtered basis function φ_(f) by the transformed signal {tilde over(y)}_(ad) to produce an error signal δ; and h) modifying at least oneconnection weight W of the neural network based on the error signal δ.